Admission of an observation-based collective forecast

Conditional admission result

The retained observation-to-future certificate bounds differences between complete histories without supplying their unknown initialization. This owner derives the contract for using one proposed carrier history as a forecast. Sections 1-4 establish its analytical admission; section 5 declares one operational test on archived synthetic readings. The research program owns its evaluation status and subsequent dependencies.

Retain the complete fine law, source class, command, support, held capacities, structural clock and sample schedule in the original declaration. Local time runs from zero at tau=200 to T=16. There are thirty vector samples, each with five mean-form coordinates, and arbitrary correlated coordinate error at most eta=1e-14. The initial common form remains unknown. Neither a zero-sum prior nor the attachment preparation may be supplied to an observation-only proposal. A common phase rotation can fix Phi_0=0; it does not fix any relative phase or winding.

1. The retained gain applies to an arbitrary sample discrepancy

Consider two exact carriers with initial mean forms bounded by X=1/100, arbitrary circular initial phases, and the same complete prescribed input. Remove the command using the original coordinates Y=U-z_cmd, Psi=Phi-H. At all readings and the future endpoint, z_cmd=0, so the observed form is Y. Let their difference be u and suppose

\[ \max_j\|u(t_j)\|_\infty\le\delta,\qquad\delta\ge0. \tag{1} \]

The static phasor construction and difference remainder do not require that delta equal the earlier fine-pair noise allowance. They use initial-form boundedness, global sine bounds, the shared command and the maximum two-unit distance from the latest sample. Their estimates are homogeneous in the carrier differences. In the notation of that proof,

\[ R_*\le A_* B_*+B_0\delta,\qquad B_*\le k(\delta+R_*)+e_0\delta+e_bB_*. \]

\[ d=1-e_b-kA_*>0,\qquad Q=\frac{k(1+B_0)+e_0}{d},\qquad B_*\le Q\delta. \tag{2} \]

Substitution into the same future residual identity gives

\[ G=w+n_0+n_bQ+(1+w)(A_*Q+B_0),\qquad \boxed{\|u(T)\|_\infty\le G\delta.} \tag{3} \]

The retained interval matrices and rational weights certify this G. Its displayed approximation 3.9317888454 is descriptive; admission uses the rebuilt exact/outward bound. Neither an assumed inverse nor a cached verdict can replace the residual and positive-denominator checks.

The comparison allows arbitrary initial edge phasors, so the two carriers need not share a winding or satisfy the original acute-gap identity. The candidate must still solve the same full-circle law. This broader comparison does not enlarge the class from which the unknown fine truth is assumed to come, or certify an NFR identity for a candidate outside that class.

2. One candidate history and its complete record residual

Let y contain the 150 admitted raw readings. Propose one exact candidate initial state (Uhat_0,Phihat_0) from y and declared model information alone, with ||Uhat_0||infinity<=X. Retain its common form and declare its phase gauge and circular representatives. Rounded proposal coordinates define the candidate actually validated; they are not exact inverses of an unknown state.

Suppose a validated numerical calculation encloses the same candidate's exact form at every reading in coordinate intervals C_jc, and its future form in intervals C_Tc about a reported center m_T, with uniform radius r_T. These enclosures must follow one uninterrupted initial state through the complete word and tail. Define the upper residual

\[ \rho=\max_{j,c}\sup_{v\in C_{jc}}|v-y_{jc}|. \tag{4} \]

For symmetric sample bounds with center m_jc and radius r_jc, this is max_(j,c)(abs(m_jc-y_jc)+r_jc). It includes historical numerical error. Intersecting a sample interval with a reading band does not prove that the exact candidate lies inside that band. Independent per-sample states likewise do not certify one jointly compatible history.

Assume the record was produced by some fine history in the original class. Its exact carrier starts at that history's actual unknown collective means. The unchanged hidden transport bound gives uniform mean-form error E_0<2.3e-18 over the entire local horizon. Rebuild E_0 from the original hidden bounds and laws; the stronger error bound from known attachment preparation does not apply here. At every reading,

\[ \|U_{truth\ carrier}(t_j)-U_{candidate}(t_j)\|_\infty \le\eta+E_0+\rho. \tag{5} \]

Apply (3), add the truth's future source error and the candidate's numerical endpoint radius, and obtain

\[ \boxed{\|U_{fine}(216)-m_T\|_\infty\le R:=G(\eta+E_0+\rho)+E_0+r_T.} \tag{6} \]

Historical numerical error is already in rho and is not added again. Only the truth needs the fine-to-carrier correction: the candidate is the exact carrier specified by its proposed initial state. A future reading with allowance eta_future has error at most R+eta_future against the forecast.

The true-form policy radius 1e-13 is admitted whenever (6) proves R<=1e-13. For example, rho<=eta+E_0 is a sufficient fitting policy if the remaining endpoint budget also satisfies this inequality. That residual cap is not necessary: the complete inequality can admit a larger residual. A future numerical protocol must fix its policy and budgets before evaluation.

3. Conditional accuracy and exact feasible witnesses

Equation (6) is conditional on fine-record membership. A carrier proposal with a small residual does not establish that a fine-class source generated the record, or that the carrier itself is a fine-class feasible witness. The record might come from a different law or violate the supplied source bounds. Good fitting does not authenticate those premises.

A stronger sufficient witness has a separate admission. Require the candidate initial collective gaps and winding to satisfy the original circular class, as well as the initial form bound. Set u=z=0 and lift

\[ x_{c,j}=\widehat U_c,\qquad \theta_{c,j}=\psi_j+\widehat\Phi_c. \tag{7} \]

The invariant-carrier proof shows this is an exact fine history for the same uniform-child input. If its whole-record upper residual satisfies rho<=eta, it is a genuine fine-class feasible member and proves nonemptiness. The retained fine-future diameter then bounds its distance from every compatible fine future; add r_T when reporting a numerical endpoint center. A residual allowed up to eta+E_0 does not establish this stronger claim.

4. Record, numerical and failure contracts

An operational consumer must retain and check:

• All 150 finite signed readings, their five-coordinate node order, exact thirty sample times, observation map, units and the declared error allowance.
• The complete law, gain, support, capacity, input and clock associations; source-class assumptions; and any preparation information actually supplied.
• The proposal's exact represented initial coordinates, circular phase conventions and common gauge, with source/algorithm/budget provenance.
• One validated candidate history, its sample enclosures and future enclosure, numerical precision/backend, stage partition, residuals and failure status.
• The retained gain evidence and rebuilt source/model-error budgets. Supplied summaries, matching labels or stale bounds cannot replace these premises.

Admission precedes float conversion. Reject nonfinite or Boolean physical values and malformed records. Circle-valued phase admission and any chosen continuous lifts remain distinct; an optimizer's representatives do not authenticate a fine-state identity. Numerical integration must retain the declared clock in both carrier rows, split actual input stages and preserve the reached state across their boundaries.

Malformed input is rejection. A failed proposal, unsupported candidate domain, inconclusive trajectory enclosure or exceeded numerical budget is unavailable candidate evidence. A failed residual or radius bound rejects that candidate for the declared policy. None establishes that every candidate fails or that the raw record is incompatible with the model. A successful bound remains conditional on its stated truth and numerical premises.

The shared prediction-transfer arithmetic and independent budget controls implement the residual/error combination, including invalid domains. That arithmetic consumes supplied enclosures and a supplied gain; it does not certify their trajectories, provenance, record membership or model identity. The shared prescribed-time integrator and retained evidence readers remain the applicable owners for those obligations. No new nonlinear trajectory, raw response or physical experiment is evaluated by this analytical result.

5. Fixed record-only software protocol

This protocol fixes one synthetic record, one deterministic proposal and one validated candidate history before extraction or numerical evaluation. Its purpose is to make the conditional forecast operational without supplying generating-state coordinates. The historical archive and its future are already public: this is a software demonstration, not a reserved response, independent validation, law comparison or physical experiment.

Record and prior evidence

The read-only input owner pins the following complete archive associations, including their attempt, result, source-tree and source-inventory hashes:

Input	Frozen producer revision	SHA-256 of response.zip
Known-preparation carrier history	8f5a28c58f23ab1586cc91ac36d49e6ddd9bbce0	c4fca529593dbfdf445d314508b78f3f188cd3fa3dce3388fa52df3a1581e4d3
Static observation certificate	de5b500f70b1626d75f12484af0e6381fc9d93af	49febee9897b4acf738d803d37128d5e9652a19e59cb082c284de780a66e3592

Re-admit the thirty absolute sample times and the first five coordinates of the original observed branch's endpoint centers, beginning with its initial center at tau=200. Use all 150 signed rational readings in time-major, child-order 0..4; do not round, add noise or select a subset. The last reading is at tau=214.00015. The command form is zero at these times, so the archived Y centers observe U. Rebuild the original source correction and carried metric radii through that last reading and require their sum to be at most eta=1e-14 at every sample. Retain extraction errors and endpoint associations as provenance, outside the proposal interface.

Rebuild the static observation/future matrices from all 2,145 retained coefficient steps, then reconstruct the dual residuals, positive bootstrap margin, G and containing-class E_0. The saved verdict or scalar gain cannot replace this arithmetic. These readers retain the original smooth-field, Taylor-derivative and source proof premises. Digests associate contents; they do not authenticate execution or prove arbitrary saved derivatives. No original coefficient or response producer is rerun.

One untrusted numerical proposal

The proposal owner receives only the thirty times, 150 readings and static 150 x 15 observation matrix. The law, command and clock are declared model information. Source phases, source initialization, extraction errors, truth labels and future values are not inference arguments. The fixed algorithm is:

1. Solve the midpoint linear design with NumPy linalg.lstsq, rcond=None, requiring rank 15. Initialize oriented edge angles with atan2 of its formal sine/cosine coefficients. Fix Phi_0=0 and obtain the other four phase representatives by cumulative sums of the first four edge angles. Initialize forms from the first five readings clipped to binary64 [-.01,.01], preserving the unknown common form.
2. Fit five forms and four phase representatives with one SciPy least_squares call: method trf, max_nfev=80, ftol=None, gtol=None, xtol=1e-13, x_scale=(.01,.01,.01,.01,.01,1,1,1,1) and residual scale 1e-4. Form bounds are binary64 [-.01,.01]; phase lifts are unbounded. Require optimizer success. Cache paired residual/Jacobian requests at identical parameters; there is no restart or alternative initializer.
3. Each trial carries the full ten-coordinate command-removed sine state and its analytic 10 x 9 sensitivities. Use DOP853 with rtol=1e-11, atol=1e-17, max_step=1/8, splitting every prescribed input stage. Integrate only the 15 plateaus and first 14 ramps, stopping at the last reading. Both rows retain the same structural clock; no state coordinate is reset at a stage boundary. This is a detached numerical proposal, not validated trajectory evidence.
4. Convert the returned finite binary64 coordinates to exact represented rationals. Clip the five forms again to the exact interval [-1/100,1/100] and set initial Phi_0=0 exactly. This represented state is the sole candidate. Retain the numerical start, optimizer status, evaluation and forward-call counts, diagnostic residual and numerical backend metadata. Backend metadata includes NumPy/SciPy versions, compiler/build architecture, BLAS/LAPACK configuration and integer widths, and available SIMD features; omit machine paths, commands and environment values.

One validated forecast and decision

The candidate consumer re-admits the record and exact candidate before integration. Run one carried candidate from tau=200 to 216 through the complete word and released tail, using the shared prescribed-time metric Taylor kernel, order 16, outward 128-bit dyadic intervals, identity metric and growth bound g=1/3000. Split at every stage and use maximum step 1/8: 143 successful steps cover the fixed horizon. The initial candidate radius is zero and the propagated numerical radius may not exceed 1e-15. Keep every compact tube, polynomial, remainder, growth and endpoint witness. No refinement or retry follows failure.

Construct all sample enclosures from this single fresh history and calculate the upper residual (4), then (6) with the rebuilt G, containing-class E_0 and eta=1e-14. Admit only if the trajectory is complete and the true-form radius is at most 1e-13. Report the five endpoint centers with their true-form bounds and separate future-reading bounds enlarged by eta. Do not inspect the archived future as an acceptance criterion. The stronger exact feasible witness in section 3 is not part of this protocol.

Failure of proposal, enclosure or policy yields explicit unavailability without a new candidate, budget, tolerance or source choice. Malformed inputs reject before use; unexpected software errors retain an incomplete attempt. A successful case establishes one operational conditional forecast, not optimizer convergence for every admissible record, source-class membership for arbitrary data, or a uniquely justified evolution law.

Run the single fixed instrument only after committing this declaration and all consumed source. Its shared exclusive assessment lifecycle records the source inventory and attempt before extraction; exact record, proposal and validated response follow in the result. Archive those bytes and receipt for read-only verification.

Retained operational software result

The single evaluation of declaration/source 53917ac7d05c4fd9c79c8ce37ed1c95c25587f50 returned conditional_record_only_forecast_certified. Its attempt, exact response archive and receipt retain the extracted record, input associations, represented candidate, proposal backend and all 143 validated steps. No old producer was replayed.

Quantity	Evaluated value, displayed approximately
Historical readings / optimizer forward calls	150 / 3
Maximum archived extraction allowance	1.489381e-21, below eta=1e-14
Whole-record validated residual rho	2.827128e-19
Rebuilt carrier gain G	3.9317888454
Containing-class source bound E_0	2.164623e-18
Candidate endpoint numerical radius r_T	9.775683e-24
Certified true-form radius R	3.932968e-14, below 1e-13
Certified future-reading radius R+eta	4.932968e-14

The five forecast centers at tau=216, in child order and structural form units, are approximately (3.256469903e-6, 1.645762162e-6, -1.253943082e-7, -1.973615505e-6, -2.803222251e-6). Use the exact archived centers and outward bounds for calculations. The binary64 optimizer's residual is only a diagnostic; the policy decision uses the separate validated history's upper residual and exact transfer arithmetic.

This establishes a computable conditional forecast for the selected record without privileged generating-state coordinates or future response. The first five raw readings initialize the form proposal; generating phases are not supplied. All retained historical enclosures belong to one candidate. It does not establish global optimizer convergence, exact fine-feasible membership, or an independently observed prediction: the record is extracted from an already published synthetic trajectory. The assumed law, source class, input and ideal clock still carry the scientific interpretation.

The read-only evidence audit rebuilds source/gain inputs, candidate/sample associations, compact radius propagation and the forecast bounds without fitting or integrating another trajectory. Retained Picard/Taylor witnesses keep their source proof premises. This forecast does not resolve whether unknown initialization can mask a changed constitutive law. That separate question needs joint law/state admission; tighter forecast arithmetic alone cannot supply it. The execution plan alone selects further investigation.